# A Peridynamic Differential Operator-Based Model for Quantifying Spatial Non-Local Transport Behavior of Pollutants in Heterogeneous Media

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## Abstract

**:**

## 1. Introduction

## 2. Model Development

#### 2.1. A Brief Review of PDDO

#### 2.2. PDDO-Based Model

#### 2.3. Numerical Algorithm of the PDDO-Based Model

## 3. Results

#### 3.1. Analytical Solution of PDDO Model

#### 3.2. One-Dimensional Results of the PDDO-Based Model

#### 3.3. Two-Dimensional Results of the PDDO-Based Model

#### 3.4. Applications

## 4. Discussion

#### Diffusion Regime of the PDDO-Based Model

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Goodarzi, D.; Mohammadian, A.; Pearson, J.; Abolfathi, S. Numerical modelling of hydraulic efficiency and pollution transport in waste stabilization ponds. Ecol. Eng.
**2022**, 182, 106702. [Google Scholar] [CrossRef] - Allen, D.T.; Cohen, Y.; Kaplan, I.R. Intermedia Pollutant Transport: Modeling and Field Measurements; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Goodarzi, D.; Abolfathi, S.; Borzooei, S. Modelling solute transport in water disinfection systems: Effects of temperature gradient on the hydraulic and disinfection efficiency of serpentine chlorine contact tanks. J. Water Process. Eng.
**2020**, 37, 101411. [Google Scholar] [CrossRef] - Qi, Y.; Zhou, P.; Wang, J.; Ma, Y.; Wu, J.; Su, C. Groundwater Pollution Model and Diffusion Law in Ordovician Limestone Aquifer Owe to Abandoned Red Mud Tailing Pit. Water
**2022**, 14, 1472. [Google Scholar] [CrossRef] - Ghiasi, B.; Noori, R.; Sheikhian, H.; Zeynolabedin, A.; Sun, Y.; Jun, C.; Hamouda, M.; Bateni, S.M.; Abolfathi, S. Uncertainty quantification of granular computing-neural network model for prediction of pollutant longitudinal dispersion coefficient in aquatic streams. Sci. Rep.
**2022**, 12, 4610. [Google Scholar] [CrossRef] - Valari, M.; Menut, L. Transferring the heterogeneity of surface emissions to variability in pollutant concentrations over urban areas through a chemistry-transport model. Atmos. Environ.
**2010**, 44, 3229–3238. [Google Scholar] [CrossRef] - Ho, Q.N.; Fettweis, M.; Spencer, K.L.; Lee, B.J. Flocculation with heterogeneous composition in water environments: A review. Water Res.
**2022**, 213, 118147. [Google Scholar] [CrossRef] - Kachiashvili, K.; Gordeziani, D.; Melikdzhanian, D. Mathematical models of Pollutants Transport with Allowance for Many Affecting Pollution Sources. Proceeding of the Urban Drainage Modeling Symposium, Orlando, FL, USA, 20–24 May 2001; pp. 692–702. [Google Scholar]
- Zhang, Y.; Sun, H.; Stowell, H.H.; Zayernouri, M.; Hansen, S.E. A review of applications of fractional calculus in Earth system dynamics. Chaos Solitons Fractals
**2017**, 102, 29–46. [Google Scholar] [CrossRef] - Mainardi, F. Fractional calculus: Theory and applications. Mathematics
**2018**, 6, 145. [Google Scholar] [CrossRef] [Green Version] - Zhang, H.; Ke, S.; Zhang, S.; Shao, J.; Chen, H. Reactive transport modeling of pollutants in heterogeneous layered paddy soils: A) Cadmium migration and vertical distributions. J. Contam. Hydrol.
**2020**, 235, 103735. [Google Scholar] [CrossRef] - Chrysikopoulos, C.V.; Kitanidis, P.K.; Roberts, P.V. Analysis of one-dimensional solute transport through porous media with spatially variable retardation factor. Water Resour. Res.
**1990**, 26, 437–446. [Google Scholar] [CrossRef] - Chrysikopoulos, C.V.; Kitanidis, P.K.; Roberts, P.V. Macrodispersion of sorbing solutes in heterogeneous porous formations with spatially periodic retardation factor and velocity field. Water Resour. Res.
**1992**, 28, 1517–1529. [Google Scholar] [CrossRef] - Chrysikopoulos, C.V.; Kitanidis, P.K.; Roberts, P.V. Generalized Taylor-Aris moment analysis of the transport of sorbing solutes through porous media with spatially-periodic retardation factor. Transp. Porous Media
**1992**, 7, 163–185. [Google Scholar] [CrossRef] - Katzourakis, V.E.; Chrysikopoulos, C.V. Impact of spatially variable collision efficiency on the transport of biocolloids in geochemically heterogeneous porous media. Water Resour. Res.
**2018**, 54, 3841–3862. [Google Scholar] [CrossRef] - Chakraborty, J.; Das, S. Molecular perspectives and recent advances in microbial remediation of persistent organic pollutants. Environ. Sci. Pollut. Res.
**2016**, 23, 16883–16903. [Google Scholar] [CrossRef] [PubMed] - Yin, M.; Zhang, Y.; Ma, R.; Tick, G.R.; Bianchi, M.; Zheng, C.; Wei, W.; Wei, S.; Liu, X. Super-diffusion affected by hydrofacies mean length and source geometry in alluvial settings. J. Hydrol.
**2020**, 582, 124515. [Google Scholar] [CrossRef] - Sun, L.; Qiu, H.; Wu, C.; Niu, J.; Hu, B.X. A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water. Wiley Interdiscip. Rev. Water
**2020**, 7, e1448. [Google Scholar] [CrossRef] - Zilitinkevich, S. Non-local turbulent transport: Pollution dispersion aspects of coherent structure of connective flows. WIT Trans. Ecol. Environ.
**1970**, 9, 53–60. [Google Scholar] - Mihailović, D.T.; Alapaty, K.; Sakradžija, M. Development of a nonlocal convective mixing scheme with varying upward mixing rates for use in air quality and chemical transport models. Environ. Sci. Pollut. Res. Int.
**2008**, 15, 296–302. [Google Scholar] [CrossRef] - Zhang, Y.; Zhou, D.; Yin, M.; Sun, H.; Wei, W.; Li, S.; Zheng, C. Nonlocal transport models for capturing solute transport in one-dimensional sand columns: Model review, applicability, limitations and improvement. Hydrol. Process.
**2020**, 34, 5104–5122. [Google Scholar] [CrossRef] - Masciopinto, C.; Passarella, G.; Caputo, M.C.; Masciale, R.; De Carlo, L. Hydrogeological Models of Water Flow and Pollutant Transport in Karstic and Fractured Reservoirs. Water Resour. Res.
**2021**, 57, e2021WR029969. [Google Scholar] [CrossRef] - Madenci, E.; Barut, A.; Futch, M. Peridynamic differential operator and its applications. Comput. Methods Appl. Mech. Eng.
**2016**, 304, 408–451. [Google Scholar] [CrossRef] - Madenci, E.; Barut, A.; Dorduncu, M. Peridynamic Differential Operator for Numerical Analysis; Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar]
- Silling, S.A. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids
**2000**, 48, 175–209. [Google Scholar] [CrossRef] [Green Version] - Madenci, E.; Roy, P.; Behera, D. Advances in Peridynamics; Spring: Tucson, AZ, USA, 2022. [Google Scholar]
- Clark, M.M. Transport Modeling for Environmental Engineers and Scientists; John Wiley & Sons: Hoboken, NJ, USA, 2011. [Google Scholar]
- Draxler, R.R.; Taylor, A.D. Horizontal dispersion parameters for long-range transport modeling. J. Appl. Meteorol. Climatol.
**1982**, 21, 367–372. [Google Scholar] [CrossRef] [Green Version] - Nezhad, M.M.; Javadi, A.A. Stochastic finite-element approach to quantify and reduce uncertainty in pollutant transport modeling. J. Hazard. Toxic Radioact. Waste
**2011**, 15, 208–215. [Google Scholar] [CrossRef] - Chen, J.; Zhuang, B.; Chen, Y.; Cui, B. Diffusion control for a tempered anomalous diffusion system using fractional-order PI controllers. ISA Trans.
**2018**, 82, 94–106. [Google Scholar] [CrossRef] [PubMed] - Sun, H.; Li, Z.; Zhang, Y.; Chen, W. Fractional and fractal derivative models for transient anomalous diffusion: Model comparison. Chaos Solitons Fractals
**2017**, 102, 346–353. [Google Scholar] [CrossRef] - Egan, B.A.; Mahoney, J.R. Numerical modeling of advection and diffusion of urban area source pollutants. J. Appl. Meteorol. Climatol.
**1972**, 11, 312–322. [Google Scholar] [CrossRef] [Green Version] - Union, J.I.G. Advection diffusion equation models in near-surface geophysical and environmental sciences. J. Ind. Geophys. Union
**2013**, 17, 117–127. [Google Scholar] - Buske, D.; Vilhena, M.; Moreira, D.; Tirabassi, T. An analytical solution of the advection-diffusion equation considering non-local turbulence closure. Environ. Fluid Mech.
**2007**, 7, 43–54. [Google Scholar] [CrossRef] - Farhane, M.; Alehyane, O.; Souhar, O. Three-dimensional analytical solution of the advection-diffusion equation for air pollution dispersion. Anziam J.
**2022**, 64, 40–53. [Google Scholar] - Laaouaoucha, D.; Farhane, M.; Essaouini, M.; Souhar, O. Analytical model for the two-dimensional advection-diffusion equation with the logarithmic wind profile in unstable conditions. Int. J. Environ. Sci. Technol.
**2022**, 19, 6825–6832. [Google Scholar] [CrossRef] - Li, Z.; Sun, H.; Sibatov, R.T. An investigation on continuous time random walk model for bedload transport. Fract. Calc. Appl. Anal.
**2019**, 22, 1480–1501. [Google Scholar] [CrossRef] - Hilfer, R. Fractional diffusion based on Riemann-Liouville fractional derivatives. J. Phys. Chem. B
**2000**, 104, 3914–3917. [Google Scholar] [CrossRef] [Green Version] - Gu, X.; Zhang, Q.; Madenci, E. Refined bond-based peridynamics for thermal diffusion. Eng. Comput.
**2019**, 36, 2557–2587. [Google Scholar] [CrossRef] - Govindaraju, R.S.; Das, B.S. Moment Analysis for Subsurface Hydrologic Applications; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2007. [Google Scholar]

**Figure 1.**Conceptual map of preference flow path in heterogeneous media, the preferential flow path is drawn with red arrows.

**Figure 2.**Conceptual map of the PDDO-based model and the non-local transport behavior (early arrivals and trailing edges) of pollutants in the framework of the PDDO-based model. Snapshots at T0 (start time), T1 and T2 times, respectively.

**Figure 3.**The weight function $w\left(\right|\xi \left|\right)$ (24) with different parameter k, exponential weight function is also drawn for comparison. All parameters are dimensionless here.

**Figure 4.**The simulation results of the one-dimensional PDDO-based model (9) with different parameter k. The weight functions are set as Equation (24). (

**a**) The breakthrough curves with parameters $R=1$, $v=0.3$ m/min, $D=0.3$ m${}^{2}$/min, ${H}_{x}=0.6$$\Omega $, (

**b**) $R=1$, $v=0.3$ m/min, $D=0.2$ m${}^{2}$/min, ${H}_{x}=0.12$$\Omega $, and $\Omega $ is the entire computing domain. The results of the advection–diffusion equation model (1) are also drawn for comparison.

**Figure 5.**The simulation results of the two-dimensional PDDO-based model (25) with different horizon ${H}_{x}$. The weight functions set as $k=3$ for Equation (24), other parameters are set as $R=1$, $v=0.3$ m/s, $D=0.3$ m${}^{2}$/s.

**Figure 6.**Comparison between the documented snapshots (symbols) and the best-fit results using the ADE (1) and PDDO-based (9) models at four times (t = 27, 132, 224, and 328 days) along the 300 m-long heterogeneous media. The weight function set as $k=2$ for Equation (24), and the horizon set as ${H}_{x}=n\xb7\Delta x$. (

**a**) 27 days, (

**b**) 132 days, (

**c**) 224 days, and (

**d**) 328 days.

**Table 1.**The best-fit parameters for F-ADE (${v}_{1}$, ${D}_{1}$, ${R}_{1}$, $\beta $), ADE (${v}_{2}$, ${D}_{2}$, ${R}_{2}$) and PDDO-based (${v}_{2}$, ${D}_{2}$, ${R}_{3}$, ${H}_{x}$) models.

Time (Days) | ${\mathit{v}}_{1}$ (m/day) | ${\mathit{D}}_{1}$ (m${}^{\mathit{\beta}}$/day) | $\mathit{\beta}$ | ${\mathit{R}}_{1}$ | ${\mathit{v}}_{2}$ (m/day) | ${\mathit{D}}_{2}$ (m${}^{2}$/day) | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{H}}_{\mathit{x}}\left(\mathit{n}\right)$ |
---|---|---|---|---|---|---|---|---|---|

27 | 60 | ||||||||

132 | 0.018 | 10 | 1.4 | 10 | 0.18 | 20 | 10 | 1 | 65 |

224 | 70 | ||||||||

328 | 80 |

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**MDPI and ACS Style**

Li, T.; Gu, X.; Zhang, Q.
A Peridynamic Differential Operator-Based Model for Quantifying Spatial Non-Local Transport Behavior of Pollutants in Heterogeneous Media. *Water* **2022**, *14*, 2455.
https://doi.org/10.3390/w14162455

**AMA Style**

Li T, Gu X, Zhang Q.
A Peridynamic Differential Operator-Based Model for Quantifying Spatial Non-Local Transport Behavior of Pollutants in Heterogeneous Media. *Water*. 2022; 14(16):2455.
https://doi.org/10.3390/w14162455

**Chicago/Turabian Style**

Li, Tianyi, Xin Gu, and Qing Zhang.
2022. "A Peridynamic Differential Operator-Based Model for Quantifying Spatial Non-Local Transport Behavior of Pollutants in Heterogeneous Media" *Water* 14, no. 16: 2455.
https://doi.org/10.3390/w14162455